Yes, I can also drive my 2011 535XI without the key if it's already running and no there is nothing wrong with the car. The car flashes a message covering a large portion of the cluster telling the driver that the key is not present. Really hard to miss. That message even appears when I get out of the car with the key in my pocket to get the mail. Just sounds like your wife either wasn't paying attention or did not know what the message was telling her. Glad it turned out OK.